Quasipositive Annuli (constructions of Quasipositive Knots and Links, Iv)

The modulus of quasipositivity q(K) of a knot K was introduced as a tool in the knot theory of complex plane curves, and can be applied to Legendrian knot theory in symplectic topology. It has also, however, a straightforward characterization in ordinary knot theory: q(K) is the supremum of the integers f such that the framed knot (K, f) embeds non-trivially on a fiber surface of a positive torus link. Geometric constructions show that −∞ < q(K), calculations with link polynomials that q(K) < ∞. The present paper aims to provide sharper lower bounds (by optimizing the geometry with positive plats) and more readily calculated upper bounds (by modifying known link polynomials), and so to compute q(K) for various classes of knots, such as positive closed braids (for which q(K) = μ(K)−1) and most positive pretzels. As an aside, it is noted that a recent result of Kronheimer & Mrowka implies that q(K) < 0 if K is slice.